The sign expresses the square root of any quantity to which it is prefixed ; thus ,/ 25 signifies the square root of 25, or 5, because 5 x5 is 25 ; and ,/ (a 6) denotes the square root of a b; and denotes thesquare root of ab+bc --, or of the quantity arising from the division of a b + 6 c by d; but ./(a b+b c) d, which has the separating tine drawn under ,/, signifies that the square root of a b+b c is to be first ta ken, and afterwards divided by d; ;so that if a were 2, b 6, c 4, and d 9, b+b c) 6 d would be but 9 (a b bed ( ) or / 4, which is 2. The sign ,/ with a figure over it is used to express the cubic er biquadratic root, &c. of any quantity; thus 64 represents the cube root of 64, or 4, because 4x4x4 is 64; and .0./ (a b+ c d) the cube root of a b+c d. In like manner de notes the root of 16, or 2,beeause 2x2X2X2 is 16, anh :/ (a b+c d) denotes the biquadra tic root of a b + c (It and so of others. Quantities thus expressed are called ra dical quantities, or surds ; of which those, consisting of one term only, as %/ a and ,/ (a b', are called simple surds; and those consisting of several terms or num bers,as ,/ (a" b') and p/ (a' b+bc) arc denominated compound surds. Ano ther commodious method of expressing radical quantities is that kvhich denotes the root by a vulgar fraction, placed at the end of a line drawn over the quantity given. In this notation, the square root is expressed by the cube root by 4, the biquadratic root by +, &c. Thus a 4 expresses the same quantity with / a, i. e. the square root of a, and (a.+a b) the same as (a.+a b), i. e. the cube root of a'-}-a b ; and denotes the cube root of the square of a, or the square of the cube root of a; and (a+ z);... the seventh power of the biquadratic root of a-1-z ; and so of others; (a') j is a, is &c. Quantities that have no ra dical sign (,/) or index annexed to them, arc called rational quantities. The sign =, called the sign of equality, signifies that the quantities between which it oc curs arc equal. Thus 2+3 = 5, shews that 2 plus 3 is equal to 5; and x=a--6 shews that 27 is equal to the difference of a and h. The mark : : signifies that the quantities between which it stands are proportional. As a : b c : ddenotes that a is in the same proportion to b as c is to d; or that if a be twice, thrice, or four times, &c. as great as b, c will be twice, thrice, or four times, &c. as great as d. When anyquantityis to be taken more than once, the number, which shows how many times it is to be taken, must be prefixed; thus 5 a denotes that the quantity a is to be taken 5 times, and 3 b c represents three times b c, and 7 ,/ (a' x 6') denotes that ,/ (a'+b') is to be taken 7 times, &c. The numbers thus prefixed are call
ed coefficients ; and if a quantity have no co-efficient, unit is understood, and it is to be taken only once. Similar or like quantities are those that are expressed by the same letters under the same pow ers, or which differ only in their co-effici ents; thus, 3 b c, 5 b c, and 8 b c, arc like quantities, and so are the radicals and 7 But un a a like quantities are those which are ex pressed by different letters, or by the same letters with different powers, as 2 a b, 5 a b., and 3 a' b. When a quantity is expressed by a single letter, orby several single letters multiplied together, without any intervening sign, as a, or 2 a b, it is called a simple quantity. But the quan tity which consists of two or more such simple quantities, connected by the signs + or , is called a compound quantity ; thus, tt - 2 a 6 + 5abcis a compound quantity ; and the simple quantities, a, 2 a I), 5 a b r, are called its terms or mem. bers. If a compound quantity consist of two terms, it is called a binorniad ; of three terms, a trinomial ; of four toms, a quadrinomial, and of many terms, a inultinomital If one of the terms of a binomial be negative, the quantity is call ed a' residual quantity. The reciprocal of any quantity is that quantity inverted, or a unity divided by it; thus 76 is the reci b 1 procal of -,and - is the reciprocal of' a a a. The letters by which any simple quantity is expressed may be ranged at pleasure, and yet retain the same signification ; thus a b and b a are the same quantity, the product of a and b being the same with that of b by a. The several terms of which any compound quantity consists may be disposed in any order at pleasure, provided they retain their proper signs. Thus a -- 2 a b 5 a' b may be written a 5 a' b 2 a b, or 2 a a' b, for all these repre sent the same thing or the quantity which remains, when from the sum of a and 5 b, the quantity 2 a b is deducted. Axioms. 1. If equal quantities be add ed to equal quantities, the stuns will be Natal.
2. If equal quantities be taken from :aqual quantities, the remainders will be equal 3. If equal quantities be multiplied. by :he same or equal quantities, the pro lucts will be equal.
4. If equal quantities be divided hy the iame, or equal quantities, the quotients be equal.
5. If the same quantity be added to and subtracted from another, the value of the atter will not be altered.
6. If a quantity be both multiplied and livided by another, its value will not be altered.